The Hawking energy on the past lightcone in cosmology

Abstract

This work studies the Hawking energy in a cosmological context. The past lightcone of a point in spacetime is the natural geometric structure closely linked to cosmological observations. By slicing the past lightcone into a one-parameter family of spacelike two-surfaces, the evolution of the Hawking energy down the lightcone is studied. Strong gravitational fields may generate lightcone self-intersections and wave front singularities. We show that in the presence of swallow-tail type singularities, the Hawking energy and its variation along the null generators of the lightcone remains well-defined and subsequently discuss its positivity and monotonicity.

Applications of the Hawking Energy in Inhomogeneous Cosmology

“…defined as the integral of the pullback of the spacetime volume-form onto S. Additionally, we may wish to assign an energy to S. In the present context, a good measure for the energy associated with a lightcone slice is the Hawking energy [1,18,19], because it relates the energy enclosed by a sphere to the amount of lightbending on it, once a foliation choice is made. Therefore, using the Hawking energy provides a way to quantify the deformation of the lightcone in terms of the energy/matter content enclosed by it.…”

“…with expansion scalars θ ± and shear tensors σ ± ab of the two null congruences orthogonal to S, represented by l a and n a . One arrives at (10) from ( 9) by using the Gauss-Bonnet theorem for a 2-sphere, S 2 R dS = 8π, together with the contracted Gauss equation [1,21],…”

“…where f S = A −1 S f dS denotes the average of f over S. By assumption, θ + θ − ≤ 0 in the weak lensing regime, and thus, the second and third term is negative. Upon exploiting the positivity of E in weak lensing regime [1] in the definition (9), we arrive at a bound of the shear-term in terms of the average density:…”

“…Phenomenologically, the Hawking quasi-local energy quantifies the lightbending of the lightcone generators. In direct continuation of [1], in which its general properties for lightcones were studied, the present work provides a direct application of these concepts, comparing domains based on energy, area and matter density in section III. In section IV, energy and area are used to assign a Robertson-Walker (RW) reference slice to every constant affine parameter slice.…”

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The past lightcone of an observer in a cosmological spacetime is the unique geometric structure directly linked to observations. After general properties of the Hawking energy along slices of the past lightcone were studied in [1], the present work continues along this path by providing explicit cosmological applications of the Hawking energy associated with a lightcone. Firstly, it is shown that amongst all two-dimensional non-trapped spheres with equal area and average matter density, a shear-free matter distribution maximizes the Hawking energy for sufficiently high densities.

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